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Introduction to factoring higher degree monomials
- [Instructor] In this video we're going to dig a little bit deeper into our knowledge or our understanding of factoring. Now factoring is something that we've been doing for many years now. You can go all the way back to when you were thinking about how would I factor the number 12. Well I could write the number 12 as three times four. I could also write it as two times six. These are all legitimate factors. Or I could try to do a prime factorization of 12, where I'm trying to write it as the product of you could view it as its most basic constituents, which would be the prime numbers. And so we've done stuff like, well 12 can be expressed as two times six. Two is prime, but then six can be expressed as two times three. And so 12 could be expressed as two times two times three, which we see right over here. This is all review, and this would be a prime factorization. And we saw an analogue when we first learned it in algebra I. In algebra I, we learned things like, and sometimes this might be in a math I class or even in a pre-algebra class, you'll learn things like hey, how do I factor x squared plus six x? And you might recognize that hey, x squared could be rewritten as x times x, and six x, that really just means six times x. And so both of them have x as a factor, and so we might wanna factor that out. And so we could rewrite this entire expression as x times x plus six. What we just did is we factored out these x's that I am circling in blue. So in general, this idea of factoring, if you're thinking about numbers, you're writing one number as the product of other numbers. If you're thinking about expressions, you're writing an expression as the product of other expressions. Well now as we go a little bit more advanced into algebra, we're gonna start thinking about doing this with higher order expressions. So we've done it with just an x or just an x squared, but now we're gonna start thinking about, well what happens if we have something to the third power, fourth power, sixth power, 10th power, 100th power? But it's really the same ideas. And we could start with monomials, which is fancy word for just a single term. So let's say I had something like six x to the seventh. What are the different ways that I could factor this? Pause this video and think about it. Can I express this as the product of two other things? Well, I could rewrite this as this as being equal to two x to the third times what? Well let's see, what do I have to multiply two by to get to six? I have to multiply it by three. And what do I have to multiply x to the third by to get to x to the seventh? I could multiply it times three x to the fourth. Notice, two times three is six, x to the third times x to the fourth is x to the seventh. We add exponents when we're multiplying, when we're multiplying things with the same base. But this isn't the only way to factor. Just as we saw that three times four wasn't the only way to factor 12. You could also express this as maybe being equal to x to the sixth times what? Well, we would still have to multiply by six then, and then we'd have to multiply by another x. So we could write this as x to the sixth times six x. So there is oftentimes multiple ways to factor a higher degree monomial like this. And there is also an analogue to doing something like a prime factorization. When you're trying to really decompose, rewrite this expression as a product of it simplest parts. How would you do that for six x to the seventh? Well, you could rewrite that. You could say six x to the seventh, well that's equal to, we could think about the six first. We know that the prime factorization of six is two times three, two times three. And then x to the seventh is just seven x's multiplied by each other. So times x times x times x times x times x, how many is that? That's five, six, and seven. And so some of what we were doing, when I said two x to the third, what we really thought about is okay, I had a two, and then I had x times x times x. And then what do I have to multiply that? Well I have to multiply that by three x to the fourth. And as we will see, being able to think about monomials in this way will be useful for factoring higher degree things that aren't monomials, things that are binomials, trinomials, or polynomials in general. And we'll do that in future videos.